Concept explainers
A coiled Hookean spring is stretched 10 cm when a 1.5-kg body is hung from it. Suppose instead that a 4.0-kg mass hangs from the spring and is set into vibration with an amplitude of 12 cm. Find (a) the force constant of the spring, (b) the maximum restoring force acting on the vibrating body, (c) the period of vibration, (d) the maximum speed and the maximum acceleration of the vibrating object, and (e) the speed and acceleration when the displacement is 9 cm.
(a)
The force constant of the spring when the mass of
Answer to Problem 32SP
Solution:
Explanation of Solution
Given data:
The mass of
The
Formula used:
The restoring force in a spring is expressed as,
Here,
The weight of an object is expressed as,
Here,
Explanation:
Consider the expression for restoring force in the spring for
Here,
Understand that the negative sign shows that the direction of restoring force is opposite to that of the elongation of spring. Consider the magnitude of restoring force
Consider the expression for weight of the
Here,
Understand that the weight of mass is equal to the restoring force on the
Substitute
Understand that the standard value of
Further solve as,
Conclusion:
The force constant for the spring is
(b)
The maximum restoring force acting on the vibrating body when the mass of
Answer to Problem 32SP
Solution:
Explanation of Solution
Given data:
The mass of
The
Formula used:
The restoring force in a spring is expressed as,
Here,
Explanation:
Consider the expression for restoring force on the mass
Here,
Substitute
The negative sign indicates that the restoring force acts in the direction opposite to the elongation of the spring.
Conclusion:
Therefore, themaximum restoring forceacting on the vibrating body is
(c)
The period of vibration of the vibrating body when the mass of
Answer to Problem 32SP
Solution:
Explanation of Solution
Given data:
The mass of
The
Formula used:
The formula for time period of a simple spring mass system in SHM is expressed as,
Here,
Explanation:
Consider the expression for time period of the spring mass system for the
Here,
Substitute
Conclusion:
The time period of vibration is
(d)
The maximum speed and maximum acceleration of the vibrating object when the mass of
Answer to Problem 32SP
Solution:
Explanation of Solution
Given data:
The mass of
The
Formula used:
The expression for acceleration of a mass in a spring mass system is written as,
Here,
The expression for velocity of mass in SHM at a location
Here,
Explanation:
Consider the expression for maximum acceleration of the mass
Here,
Substitute
The negative sign shows that the acceleration is in the direction opposite to elongation. Therefore, the maximum acceleration is
Consider the expression for velocity of the mass
Here,
Understand that the velocity is maximum at equilibrium position. Therefore, substitute 0 for
Conclusion:
The maximum acceleration is
(e)
The speed and acceleration of the vibrating object when displacement is
Answer to Problem 32SP
Solution: The acceleration when displacement is
Explanation of Solution
Given data:
The mass of
The
Formula used:
The expression for acceleration of a mass in a spring mass system is written as,
Here,
The expression for velocity of mass in SHM at a location
Here,
Explanation:
Consider the expression for acceleration of the mass
Here,
Substitute
The negative sign shows that the acceleration is in the direction opposite to elongation. Therefore, the acceleration when displacement is
Consider the expression for velocity of the mass
Here,
Substitute
The velocity when displacement is
Conclusion:
The acceleration when displacement is
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Chapter 11 Solutions
Schaum's Outline of College Physics, Twelfth Edition (Schaum's Outlines)
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